David Cameron's position on Europe is not just good for the UK, it is also good for the EU. In his much-awaited speech the Prime Minister set out a clear-eyed vision of the benefits of European integration but an equally acute analysis of the weaknesses of the current EU.

Most importantly, the speech laid out a process for regaining the British people's trust in the UK's relationship with the EU. If we stay inside the bloc it will be because Britons want to stay. If we leave, it will be by popular consent.

But the Prime Minister's offer to the UK is also an opportunity for other Europeans: the current EU does not – indeed, it cannot – serve the best interests of Europe. The recasting of the union can work in the interests of all Europeans.

An open, flexible and dynamic Europe will help the UK along with other European countries grow, and prosper. A closed, rigid, over-regulated and centralised Europe will help nobody. The key now is to persuade our European allies of this.

To achieve this, Britain must undertake a diplomatic campaign to persuade our allies that Britain is not only looking out for itself but offers a way forward that will strengthen the entire Continent - democratically, economically and internationally.

First, we have to create a debate across the continent about how Europe retains its economic advantages. The talk of growth has been hijacked by those who think prosperity comes from a bigger EU budget rather than a bigger, freer market.

The UK has to lead a campaign in key countries about the future private-sector sources of European prosperity. One way to do so may be to commission a group of economists, diplomats and business leaders to give a clear-eyed and evidence-based assessment of future economic opportunities and areas of co-operation for the EU. Crucially, it should also set out areas where the EU should disengage.

This is a good moment to seize the agenda. In 2014 every member of the EU will hold elections to the European Parliament, and a new set of leaders will be appointed – a new European Commission president, a new President of the European Council and a new Secretary-General of NATO.

The people that will lead the Union from early 2015 should do so with an agenda that is partly shaped in London. That means work needs to begin now in every possible area of co-operation - from energy policy, and telecommunications, to competition and trade, and development and climate change. Leaving the agenda-setting to the European bureaucracy will not benefit Britain nor, indeed, the long-term prospects of the EU or its individual members.

A renewed move to liberalise Europe's market in services should be a core part of this new agenda. Europe missed a chance a few years ago to create a genuinely open market where professionals and service workers, from accountants to plumbers and hairdressers, can freely offer their services anywhere in Europe. As a result national governments often hide behind complex guidelines on how to screen national regulations, failing to remove costly obstacles to commerce. The outcome: Europeans buy nine-tenths of their services from firms in their home countries and Britain's competition-beating companies are denied the access to market that they need to spur their own and national growth.

The second task is diplomatic. After years of Labour running down the Foreign Office, William Hague, the foreign secretary, has rebuilt relationships with old allies like Australia and new friends such as Brazil. But a European challenge remains.

In France, the governing Socialist party is almost implacably hostile to Britain, despite the good Franco-British cooperation over Libya and Mali. Leading figures in Poland have gone from seeing Britain as a home in times of trouble to an obstacle to Poland's own ambitions. And in Spain, one often hears that it has been years since a British leader paid an official visit. Even in Germany, despite David Cameron’s relationship-building with Chancellor Merkel, many perceive the UK as unhelpful.

If we are to succeed in changing Europe we have to build multi-layered links with these nations - not just at a diplomatic level but between politicians, academics and opinion-formers. Much as the United States runs a Visitor's Programme, targeting up-and-coming leaders, inviting them to America and displaying the country in its richness, so Britain needs to do the same on the continent. We have to win over the next generation of European leaders.

We must also find ways to replicate elements of the French-German co-operation with other states. For example we could organise a joint sitting of the UK and Polish Parliaments and a joint UK/Dutch cabinet meeting. These events may never have the historical force of the French-German cooperation, but could be valuable nonetheless.

Some may see such initiatives as mere form, but they are not; difficult negotiations over the future of the EU – and Britain's relationship with the bloc – will require a denser range of links not just with other governments, but with the opinion-formers that influence them. As a corollary, Britain's diplomats need to think less about persuading colleagues in other foreign ministries and look for ways to generate a public debate where they serve.

The decrease in the EU budget has shown that Britain is not isolated diplomatically, but has European friends who share our vision. We now need to build on this success with a thoughtful and innovative campaign, which sets the stage not only for Britain's new relationship with the EU but also for a new phase for Europe.